Editing Del (section)

==Notational uses==
Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the [[gradient]], [[divergence]], [[Curl (mathematics)|curl]], [[directional derivative]], and [[Laplacian]].

===Gradient===
The vector derivative of a [[scalar field]] <math>f</math> is called the [[gradient]], and it can be represented as:
: <math>\operatorname{grad}f = {\partial f \over \partial x} \hat\mathbf x + {\partial f \over \partial y} \hat\mathbf y + {\partial f \over \partial z} \hat\mathbf z=\nabla f</math>

It always points in the [[Direction_(geometry)|direction]] of greatest increase of <math>f</math>, and it has a [[magnitude (mathematics)|magnitude]] equal to the maximum rate of increase at the point&mdash;just like a standard derivative. In particular, if a hill is defined as a height function over a plane <math>h(x,y)</math>, the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.

In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:
: <math>\nabla(f g) = f \nabla g + g \nabla f</math>

However, the rules for [[dot product]]s do not turn out to be simple, as illustrated by:
: <math>\nabla (\mathbf u \cdot \mathbf v) = (\mathbf u \cdot \nabla) \mathbf v + (\mathbf v \cdot \nabla) \mathbf u + \mathbf u \times (\nabla \times \mathbf v) + \mathbf v \times (\nabla \times \mathbf u)</math>

===Divergence===
The [[divergence]] of a [[vector field]]
<math> \mathbf v(x, y, z) = v_x \hat\mathbf x  + v_y \hat\mathbf y + v_z \hat\mathbf z </math> is a [[scalar field]] that can be represented as:
:<math>\operatorname{div}\mathbf v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z} = \nabla \cdot \mathbf v </math>

The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point.

The power of the del notation is shown by the following product rule:
:<math> \nabla \cdot (f \mathbf v) = (\nabla f) \cdot \mathbf v + f (\nabla \cdot \mathbf v) </math>

The formula for the [[vector product]] is slightly less intuitive, because this product is not commutative:
:<math> \nabla \cdot (\mathbf u \times \mathbf v) = (\nabla \times \mathbf u) \cdot \mathbf v - \mathbf u \cdot (\nabla \times \mathbf v)</math>

===Curl===
The [[Curl (mathematics)|curl]] of a vector field <math>\mathbf v(x, y, z) = v_x\hat\mathbf x  + v_y\hat\mathbf y + v_z\hat\mathbf z</math> is a [[vector field|vector]] function that can be represented as:
:<math>\operatorname{curl}\mathbf v = \left({\partial v_z \over \partial y} - {\partial v_y \over \partial z} \right) \hat\mathbf x + \left({\partial v_x \over \partial z} - {\partial v_z \over \partial x} \right) \hat\mathbf y + \left({\partial v_y \over \partial x} - {\partial v_x \over \partial y} \right) \hat\mathbf z = \nabla \times \mathbf v</math>

The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centered at that point.

The vector product operation can be visualized as a pseudo-[[determinant]]:
:<math>\nabla \times \mathbf v = \left|\begin{matrix} \hat\mathbf x & \hat\mathbf y & \hat\mathbf z \\[2pt] {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\[2pt] v_x & v_y & v_z \end{matrix}\right|</math>

Again the power of the notation is shown by the product rule:
:<math>\nabla \times (f \mathbf v) = (\nabla f) \times \mathbf v + f (\nabla \times \mathbf v)</math>

The rule for the vector product does not turn out to be simple:
:<math>\nabla \times (\mathbf u \times \mathbf v) = \mathbf u \, (\nabla \cdot \mathbf v) - \mathbf v \, (\nabla \cdot \mathbf u) + (\mathbf v \cdot \nabla) \, \mathbf u - (\mathbf u \cdot \nabla) \, \mathbf v</math>

===Directional derivative===
The [[directional derivative]] of a scalar field <math>f(x,y,z)</math> in the direction
<math>\mathbf a(x,y,z) = a_x \hat\mathbf x + a_y \hat\mathbf y + a_z \hat\mathbf z </math> is defined as:

:<math>(\mathbf a\cdot\nabla)f=\lim_{h \to 0}{\frac{f(x+a_xh,y+a_yh,z+a_zh) - f(x,y,z)}{h}}.</math>
Which is equal to the following when the gradient exists
:<math>\mathbf a\cdot\operatorname{grad}f = a_x {\partial f \over \partial x} + a_y {\partial f \over \partial y} + a_z {\partial f \over \partial z} = \mathbf a \cdot (\nabla f) </math>

This gives the rate of change of a field <math>f</math> in the direction of <math>\mathbf a</math>, scaled by the magnitude of <math>\mathbf a</math>. In operator notation, the element in parentheses can be considered a single coherent unit; [[fluid dynamics]] uses this convention extensively, terming it the [[convective derivative]]&mdash;the "moving" derivative of the fluid.

Note that <math> (\mathbf a \cdot \nabla) </math> is an operator that takes scalar to a scalar. It can be extended to operate on a vector, by separately operating on each of its components.

===Laplacian===
The [[Laplace operator]] is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:
: <math>\Delta = {\partial^2 \over \partial x^2} + {\partial^2 \over \partial y^2} + {\partial^2 \over \partial z^2} = \nabla \cdot \nabla = \nabla^2</math>
and the definition for more general coordinate systems is given in [[vector Laplacian]].

The Laplacian is ubiquitous throughout modern [[mathematical physics]], appearing for example in [[Laplace's equation]], [[Poisson's equation]], the [[heat equation]], the [[wave equation]], and the [[Schrödinger equation]].

===Hessian matrix===

While <math>\nabla^2</math> usually represents the [[Laplacian]], sometimes <math>\nabla^2</math> also represents the [[Hessian matrix]]. The former refers to the inner product of <math>\nabla</math>, while the latter refers to the [[dyadic product]] of <math>\nabla</math>:

: <math>\nabla^2 = \nabla \cdot \nabla^T</math>.

So whether <math>\nabla^2</math> refers to a Laplacian or a Hessian matrix depends on the context.

===Tensor derivative===
Del can also be applied to a vector field with the result being a [[tensor]]. The [[tensor derivative]] of a vector field <math>\mathbf{v}</math> (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as <math>\nabla \otimes \mathbf{v}</math>, where <math>\otimes</math> represents the [[dyadic product]]. This quantity is equivalent to the transpose of the [[Jacobian matrix]] of the vector field with respect to space. The divergence of the vector field can then be expressed as the [[Trace (linear algebra)|trace]] of this matrix.

For a small displacement <math>\delta \mathbf{r}</math>, the change in the vector field is given by:
: <math> \delta \mathbf{v} = (\nabla \otimes \mathbf{v})^T \sdot \delta \mathbf{r} </math>